Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 058, 14 pages      math-ph/0703044
Contribution to the Vadim Kuznetsov Memorial Issue

From su(2) Gaudin Models to Integrable Tops

Matteo Petrera a and Orlando Ragnisco b
a) Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, D-85747 Garching bei München, Germany
b) Dipartimento di Fisica E. Amaldi, Università degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy

Received March 13, 2006; Published online April 20, 2007

In the present paper we derive two well-known integrable cases of rigid body dynamics (the Lagrange top and the Clebsch system) performing an algebraic contraction on the two-body Lax matrices governing the (classical) su(2) Gaudin models. The procedure preserves the linear r-matrix formulation of the ancestor models. We give the Lax representation of the resulting integrable systems in terms of su(2) Lax matrices with rational and elliptic dependencies on the spectral parameter. We finally give some results about the many-body extensions of the constructed systems.

Key words: Gaudin models; spinning tops.

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  1. Audin M., Spinning tops, Cambridge University Press, 1996.
  2. Belavin A.A., Drinfel'd V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funktsional. Anal. i Prilozhen. 16 (1982), 1-29.
  3. Belokolos E.D., Bobenko A.I., Enol'skii V.Z., Its A.R., Matveev V.B., Algebro-geometrical methods in the theory of integrable equations, Springer, Heidelberg, 1991.
  4. Bobenko A.I., Suris Yu.B., Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top, Comm. Math. Phys. 204 (1999), 147-188, solv-int/9810018.
  5. Gaudin M., Diagonalisation d'une classe d'Hamiltoniens de spin, J. Physique 37 (1976), 1089-1098.
  6. Gaudin M., La fonction d' onde de Bethe, Masson, Paris, 1983.
  7. Gavrilov L., Zhivkov A., The complex geometry of Lagrange top, L' Enseign. Math. 44 (1998), 133-170, solv-int/9809012.
  8. Hone A.N.W., Kuznetsov V.B., Ragnisco O., Bäcklund transformations for the sl(2) Gaudin magnet, J. Phys. A: Math. Gen. 34 (2001), 2477-2490, nlin.SI/0007041.
  9. Inönü E., Wigner E.P., On the contraction of groups and their representations, Proc. Natl. Acad. Sci. 39 (1953), 510-524.
  10. Jurco B., Classical Yang-Baxter equations and quantum integrable systems, J. Math. Phys. 30 (1989), 1289-1293.
  11. Kalnins E.G., Kuznetsov V.B., Miller W.Jr., Separation of variables and the XXZ Gaudin magnet, Rend. Sem. Mat. Univ. Polit. Torino 53 (1995), 109-120, hep-th/9412190.
  12. Kalnins E.G., Kuznetsov V.B., Miller W.Jr., Quadrics on complex Riemannian spaces of constant curvature, separation of variables and the Gaudin magnet, J. Math. Phys. 35 (1994), 1710-1731, hep-th/9308109.
  13. Kuznetsov V.B., Quadrics on real Riemannian spaces of constant curvature: separation of variables and connection with Gaudin magnet, J. Math. Phys. 33 (1992), 3240-3254.
  14. Kuznetsov V.B., Petrera M., Ragnisco O., Separation of variables and Bäcklund transformations for the symmetric Lagrange top, J. Phys. A: Math. Gen. 37 (2004), 8495-8512, nlin.SI/0403028.
  15. Kuznetsov V.B., Sklyanin E.K., Few remarks on Bäcklund transformations for many-body systems, J. Phys. A: Math. Gen. 31 (1998), 2241-2251, solv-int/9711010.
  16. Kuznetsov V.B., Vanhaecke P., Bäcklund transformations for finite-dimensional integrable systems: a geometric approach, J. Geom. Phys. 44 (2002), 1-40, nlin.SI/0004003.
  17. Musso F., Petrera M., Ragnisco O., Algebraic extensions of Gaudin models, J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 482-498, nlin.SI/0410016.
  18. Musso F., Petrera M., Ragnisco O., Satta G., A rigid body dynamics derived from a class of extended Gaudin models: an integrable discretization, Regul. Chaotic Dyn. 10 (2005), 363-380, math-ph/0503002.
  19. Musso F., Petrera M., Ragnisco O., Satta G., Bäcklund transformations for the rational Lagrange chain, J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 240-252, nlin.SI/0412017.
  20. Petrera M., Integrable extensions and discretizations of classical Gaudin models, PhD Thesis Dipartimento di Fisica E. Amaldi, Università degli Studi Roma Tre, Rome, Italy, 2007.
  21. Petrera M., Suris Yu.B., Integrable discretizations of rational su(2) Gaudin models and their extensions, in preparation.
  22. Reyman A.G., Semenov-Tian-Shansky M.A., Group-theoretical methods in the theory of finite-dimensional integrable systems, in Dynamical Systems VII, Editors V.I. Arnold and S.P. Novikov, Encyclopaedia of Mathematical Sciences, Vol. 16, Berlin, Springer, 1994, 116-225.
  23. Sklyanin E.K., Some algebraic structures connected with the Yang-Baxter equation, Funktsional. Anal. i Prilozhen. 16 (1982), no. 4, 27-34, 96 (in Russian).
  24. Sklyanin E.K., Separation of variables in the Gaudin model, in Differentsialnaya Geom. Gruppy Li i Mekh. IX, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 164 (1987), 151-169, 198 (English transl.: J. Soviet Math. 47 (1989), no. 2, 2473-2488).
  25. Sklyanin E.K., The Poisson structure of the classical XXZ-chain, in Vopr. Kvant. Teor. Polya i Statist. Fiz. 7, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 161 (1987), 88-97, 176-177, 179-180 (English transl.: J. Soviet Math. 46 (1989), no. 5, 2104-2111).
  26. Sklyanin E.K., Takebe T., Algebraic Bethe ansatz for XYZ Gaudin model, Phys. Lett. A 219 (1996), 217-225, q-alg/9601028.
  27. Suris Yu.B., The problem of integrable discretization: Hamiltonian approach, Progress in Mathematics, Vol. 219, Birkhäuser Verlag, Basel, 2003.
  28. Suris Yu.B., The motion of a rigid body in a quadratic potential: an integrable discretization, Int. Math. Res. Not. 12 (2000), 643-663, solv-int/9909009.
  29. Suris Yu.B., Integrable discretizations of some cases of the rigid body dynamics, J. Nonlinear Math. Phys. 8 (2001), 534-560, nlin.SI/0105012.

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